Write the equation for a parabola with a focus at $(6,-4)$ and a directrix at $y=-7$. $y=$
Explanation: The strategy A parabola is defined as the set of all points that are the same distance away from a point (the focus) and a line (the directrix). Let $(x,y)$ be a point on the parabola. Then the distance between $(x,y)$ and the focus, $(6,-4)$, is equal to the distance between $(x,y)$ and the directrix, $y=-7$. Once we find these distances, we can equate them in order to derive the equation of our parabola. Finding the distances from $(x,y)$ to the focus and the directrix The distance between $(x,y)$ and $(6,-4)$ is $\sqrt{(x-6)^2+(y+4)^2}$. [How did we find that?] Similarly, the distance between $(x,y)$ and the line $y=-7$ is $\sqrt{(y+7)^2}$. [How did we know that?] Deriving the formula by equating the distances $\begin{aligned} \sqrt{(y+7)^2} &= \sqrt{(x-6)^2+(y+4)^2} \\\\ (y+7)^2 &= (x-6)^2+(y+4)^2 \\\\ {y^2}+14y{+49} &= (x-6)^2{+y^2}{+8y}+16\\\\ 14y{-8y}&=(x-6)^2+16{-49} \\\\ 6y&=(x-6)^2-33 \\\\ y&=\dfrac{(x-6)^2}{6}-\dfrac{11}{2}\end{aligned}$ The answer The equation of our parabola is $y=\dfrac{(x-6)^2}{6}-\dfrac{11}{2}$. Here is the graph of our parabola. As expected, the distance between a point on the parabola, $(x,y)$, and the focus is the same as the distance between $(x,y)$ and the directrix. ${2}$ ${4}$ ${6}$ ${8}$ ${10}$ ${12}$ ${14}$ ${\llap{-}2}$ ${\llap{-}4}$ ${\llap{-}6}$ ${\llap{-}8}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ $y$ $x$ ${(x,y)}$